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Is the standard deviation always smaller than the sum of squares?

Published in Statistics 3 mins read

No, the standard deviation is not always smaller than the sum of squares.

The relationship between standard deviation and sum of squares is complex and depends on the data set. While a larger sum of squares generally leads to a larger standard deviation, it doesn't guarantee that the standard deviation will always be smaller. Let's break this down:

Understanding the Terms

  • Sum of Squares (SS): This is the sum of the squared deviations from the mean of a data set. It measures the total variability in the data. Mathematically, SS = Σ(xi - μ)2, where xi is each data point and μ is the mean.

  • Standard Deviation (SD): This measures the spread or dispersion of a set of values around the mean. It's the square root of the variance. Mathematically, SD = √[Σ(xi - μ)2 / (N-1)] (for sample standard deviation) or √[Σ(xi - μ)2 / N] (for population standard deviation), where N is the number of data points.

The Relationship

The standard deviation is calculated from the sum of squares. Specifically, the variance is the sum of squares divided by (N-1) for a sample or N for a population, and the standard deviation is the square root of the variance.

Because the sum of squares is divided by the number of data points (or a value close to it) and then the square root is taken, the standard deviation can be larger or smaller than the sum of squares, depending on the magnitude of the sum of squares and the number of data points.

Examples

  • Scenario 1: Standard Deviation Smaller than Sum of Squares

    Consider the data set: {1, 2, 3, 4, 5}

    • Mean (μ) = 3
    • Sum of Squares (SS) = (1-3)2 + (2-3)2 + (3-3)2 + (4-3)2 + (5-3)2 = 4 + 1 + 0 + 1 + 4 = 10
    • Sample Standard Deviation (SD) = √[10 / (5-1)] = √(10/4) = √2.5 ≈ 1.58

    In this case, the standard deviation (1.58) is smaller than the sum of squares (10).

  • Scenario 2: Standard Deviation Larger than Sum of Squares (Hypothetical, demonstrating the principle)

    This is less common with typical datasets, but conceptually, imagine a dataset where the values are very close to the mean, resulting in extremely small deviations, but a very large sample size. It's difficult to create a realistic example where SD > SS with naturally occurring data, but mathematically, it is possible. The key is that the division by N-1 or N and the square root operation can reduce the overall value considerably.

Key Takeaway

While an increase in the sum of squares usually corresponds to an increase in the standard deviation (as mentioned in the reference), there's no guarantee that the value of the standard deviation will always be smaller than the value of the sum of squares. The number of observations plays a critical role in determining the relative magnitudes.

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